"She only wants half a piece of cake."
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It's better, I think, but technically it pretty much suffers from the same problem.
I think we could use the qualifier "regular" — "a regular serving", "a regular(-sized) piece", etc. — and then say that she wants half of that, though it would require the speaker and the listener to have a shared understanding of what a regular serving/piece was.
But in context, perhaps either "piece" or "serving" on its own could be understood to mean the "regular" version of either of those.
"想" has to be followed by a verb to mean "to want (to)". In front of a noun it doesn't mean "to want" in the English sense of "to be thinking of having". Instead it means "to miss (someone or something)", "to think up (an idea)", etc. Your sentence makes it sound as though she misses the cake.